EFFECT OF SHORT-DURATION-HIGH-IMPULSE VARIABLE AXIAL AND TRANSVERSE LOADS ON REINFORCED CONCRETE COLUMN

Transcription

1 EFFECT OF SHORT-DURATION-HIGH-IMPULSE VARIABLE AXIAL AND TRANSVERSE LOADS ON REINFORCED CONCRETE COLUMN By THIEN PHUOC TRAN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 29 1

2 29 Thien Phuoc Tran 2

3 To my parents, brother, sister, my wife and all my children 3

4 ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Theodor Krauthammer, for all his valuable advices and guidance. I would also like to extend my appreciation to Dr. Serdar Astarlioglu for all his constructive ideas and his attentiveness to facilitate the completion of this thesis paper. I am grateful to the Canadian Armed Forces, particularly the Defence Research and Development Canada in Suffield for providing the opportunity for me to complete this postgraduate program. I would like to thank all my friends and colleagues in Canada and at the Center for Infrastructure Protection and Physical Security, University of Florida for all the supports during the last two years. Lastly, I am indebted to my parents, my brother, my sister, my wife and all my children for all the sacrifices that they have made to provide the opportunity for my achievement. 4

5 TABLE OF CONTENTS ACKNOWLEDGMENTS... 4 LIST OF TABLES... 7 LIST OF FIGURES... 8 LIST OF SYMBOLS AND ABBREVIATIONS ABSTRACT CHAPTER 1 INTRODUCTION page 1.1 Problem Statement Objectives and Scope Research Significance LITERATURE REVIEW Introduction Blast Loading on Structure Structure and Its Equivalent System Introduction Equivalent Mass Equivalent Load Shape Functions Resistance Functions Static Analysis Flexural Behavior Influence of Shear on Flexural Response Influence of Axial Force on Shear and Flexural Responses Direct Shear Mode of Response Dynamic Analysis Newmark-Beta Method Dynamic Resistance Functions Modified Equivalent Parameters for SDOF System Mass factor Load factor Dynamic Reactions Pressure-Impulse (P-I) Diagrams Characteristics of P-I Diagrams Derivation of P-I Diagrams Summary

6 3 METHODOLOGY Introduction Load Determination Overview of Structure Effects of Transverse Loads Axial Loads Load Deformation Analysis Computations of Dynamic Reactions, Shear and Flexural Responses Computation of Dynamic Reactions for the Supported Mass Computation of Shear and Flexural Responses on Columns Summary ANALYSIS Introduction Description of DSAS Validations with Experimental Data Experimental Data Material and physical properties Loading functions ABAQUS Validations DSAS Validations Comparison of Results from ABAQUS and DSAS to Experimental Data Validation for Beam Subject to Uniform Load Using ABAQUS and DSAS Validation for Column Using ABAQUS and DSAS Summary PARAMETRIC STUDY Description of Columns Columns Subject to Transverse and Constant Axial Load Columns Subject to Transverse, Constant and Variable Axial Loads Summary CONCLUSIONS AND RECOMMENDATIONS Summary Conclusions Recommendations APPENDIX SAMPLE ABAQUS INPUT FILE BEAM 1-C LIST OF REFERENCES BIOGRAPHICAL SKETCH

7 LIST OF TABLES Table page 2-1 Definition of direct shear-slip δ relationships Concrete material properties Steel reinforcements material properties Material model properties for beams in ABAQUS Strain rate hardening and material enhancement factors Comparison on ABAQUS, DSAS and experiment results Summary of columns physical and material properties Constant axial load cases Comparisons on displacements resulted from ABAQUS and DSAS for P1 and P Comparisons on displacements resulted from ABAQUS and DSAS for P3 and P Comparisons on displacements induced by constant and variable axial loads

8 LIST OF FIGURES Figure page 2-1 Air-burst explosion Ground-burst explosion Typical blast wave pressure time-history graph Partial and simplied time-history graph Real and typical equivalent SDOF system Mass diagram Load diagram Resistance functions Stress and strain diagram of a cross section Typical shear failure of a column Flexure-shear interaction model valley of diagonal failure Flexure-shear interaction model Direct shear-slip relationship Degrading stiffness method Idealized hysteresis loops for reinforced concrete Typical response of a SDOF system Dynamic reactions for beam with arbitrary boundary conditions Typical pressure-impulse diagram Search algorithm in developing P-I diagram Typical P-I diagram for multi failure modes Blast loads on structure Load diagram for supported mass Load diagram for column

9 3-4 Newton-Rhapson method Typical load-deformation diagram of a strucutre Spherical constant arc length criterion for SDOF system Supported mass diagram Column with axial load Flow chart for determining dynamic reactions based on failure mode Newmark-Beta method for computing displacement Beam 1-C layout Load function for beam 1-C Load function for beam 1-G Load function for beam 1-H Load function for beam 1-I Load function for beam 1-J Typical modeling of experimental beam using ABAQUS Graphical presentation of parameters in the Modified Drucker-Prager/Cap Model Stress-strain relationship tension reinforcements beam 1-C Stress-strain relationship compression reinforcement beam 1-C Stress-strain relationship tension reinforcements beam 1-G Stress-strain relationship compression reinforcement beam 1-G Stress-strain relationship tension reinforcements beam 1-H Stress-strain relationship compression reinforcement beam 1-H Stress-strain relationship tension reinforcements beam 1-I Stress-strain relationship compression reinforcement beam 1-I Stress-strain relationship tension reinforcements beam 1-J Stress-strain relationship compression reinforcement beam 1-J

10 4-19 Typical DSAS data entry screen Comparison of displacement-time history for beam 1-C Comparison of displacement-time history for beam 1-G Comparison of displacement-time history for beam 1-H Comparison of displacement-time history for beam 1-I Comparison of displacement-time history for beam 1-J Loading function for 5 pounds of Trinitrotoluene (TNT) at 2 ft Displacement-time history of beam 1-C Typical column layout in ABAQUS Loads and boundary conditions for column in ABAQUS Longitudinal and transverse steel reinforcement layout in ABAQUS Displacement-time history column subject to blast load Axial-moment interaction diagram confined conrete DSAS versus CRSI Stress-strain relationship tension and compression reinforcements column Stress-strain relationship tension and compression reinforcements column Stress-strain relationship tension and compression reinforcements column Stress-strain relationship tension and compression reinforcements column Axial-moment interaction diagram 8 No. 7 RC confined Displacement-time history diagram 8 No. 7 RC confined Flexure-resistance diagram 8 No. 7 RC confined Moment-curvature diagram 8 No. 7 RC confined Pressure-impuluse diagram 8 No. 7 RC confined Axial-moment interaction diagram 8 No. 1 RC confined Displacement-time history diagram 8 No. 1 RC confined Flexure-resistance diagram 8 No. 1 RC confined

11 5-14 Moment-curvature diagram 8 No. 1 RC confined Pressure-impuluse diagram 8 No. 1 RC confined Axial-moment interaction diagram 12 No. 11 RC confined Displacement-time history diagram 12 No. 11 RC, P = to 57 kips confined Displacement-time history diagram 12 No. 11 RC, P > 57 kips confined Flexure-resistance diagram 12 No. 11 RC confined Moment-curvature diagram 12 No. 11 RC confined Pressure-impuluse diagram 12 No. 11 RC confined Axial-moment interaction diagram 4 No. 14 RC confined Displacement-time history diagram 4 No. 14 RC confined Flexure-resistance diagram 4 No. 14 RC confined Moment-curvature diagram 4 No. 14 RC confined Pressure-impuluse diagram 4 No. 14 RC confined Variable axial load profile Displacement-time history 8 No. 7 RC P Pbal + Pvar Displacement-time history 8 No. 7 RC P > Pbal + Pvar Displacement-time history 8 No. 1 RC P Pbal + Pvar Displacement-time history 8 No. 1 RC P > Pbal + Pvar Displacement-time history 12 No. 11 RC P Pbal + Pvar Displacement-time history 12 No. 11 RC P > Pbal + Pvar Displacement-time history 4 No. 14 RC P Pbal + Pvar Displacement-time history 4 No. 14 RC P > Pbal + Pvar

12 LIST OF SYMBOLS AND ABBREVIATIONS a a A g Distance from support to the load Speed of sound Cross-section gross area ATM Atmospheric pressure d Depth from the top of concrete to the layer of reinforced bars f Specified compressive strength c f y F(t) h I I s r K K K K K L τ e L M w m M M M M e fl n Yield strength Load time function Depth of concrete cross-section Positive incident impulse Positive normal reflected impulse Tangent stiffness matrix Coefficient matrix Equivalent stiffness Load factor Mass factor Wavelength of positive phase Mass Moment Equivalent mass Ultimate moment due to pure flexure Nominal flexural strength 12

13 M M N P e P d P P P r u t u so t P u q R s R R G m (t) (t) R(t) SRF t t A pos U V c w W α ε cm Ultimate moment due to shear and flexure Total mass Factored axial force normal to cross section Equivalent load Downward load Reflected pressure Incident pressure Actual total load Upward load Dynamic pressure Stand-off distance Load vector Maximum plastic-limit load Resistance function Shear reduction factor Time of arrival of blast wave Positive duration of positive phase Shock front velocity Nominal shear strength Width of concrete beam cross-section Trinitrotoluene (TNT) equivalent charge weight Angle of incident Maximum compression strain 13

14 Δ Δu ΔR γ λ φ φ(x) ρ ρ ρ s Ψ(x) σ τ m Displacement Incremental displacement Incremental load Inertia and load proportionality factors Load multiplier Curvature Assume shape function Reinforcement ratio Air density behind the shock front Density of air beyond the blast wave at atmospheric pressure Shape function Stress Maximum shear stress 14

15 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science EFFECT OF SHORT-DURATION-HIGH-IMPULSE VARIABLE AXIAL AND TRANSVERSE LOADS ON REINFORCED CONCRETE COLUMN Chair: Theodor Krauthammer Major: Civil Engineering By Thien Phuoc Tran May 29 Previous studies were conducted on the deformations of reinforced concrete columns induced by blast load that combined both axial and transverse loading components. Most of those studies assumed that the response of the mass supported by the column in its axial direction developed much slower compared to that in the lateral movement. Thus, the load transferred from the supported mass to the column in its axial direction could be treated as a static load. Moreover, when comparing the vertical displacement with the lateral displacement of the column, it was assumed that the former was much smaller, and therefore was negligible. Consequently, the failure of the column was assumed to be governed by the flexure caused by transverse loads. While this may be true, the effect of variable axial loads may still be an important factor in determining the failure of the column. Thus, the above simplified assumption should be re-examined to determine the actual effect of variable axial loads on the behavior of a column. 15

16 CHAPTER 1 INTRODUCTION 1.1 Problem Statement Progressive collapse of a building is normally caused by an abrupt failure of one or more structural bearing members such as beams or columns. Therefore, the endurance of these members under short duration but highly impulsive loads is crucial for the survivability of the building. While beams are normally subject to transverse loads, columns are always exposed to both transverse and axial loads. In practice, it is assumed that failure of a column is normally caused by transverse rather than axial loads. This may not be accurate, particularly in the case where a structure is subjected to short duration but highly impulsive loads such as blast loads. While the failure of the column will most likely be induced by the transverse loads, the effect of variable axial loads should also be considered as a contributing factor. The column resistance may be reduced due to the variable axial loads under the same material and physical properties, and the column may fail sooner. On the other hand, the alterations in directions and the eccentricity of the variable axial loads over the time period may act as an enhancement factor to the strength of the column, thus preventing it from failing in the early stage. The discussion on eccentricity is, however, not a in the scope of this work and therefore is not included in this study. 1.2 Objective and Scope The objective of this research is to determine the actual effect of variable axial loads on the column, allowing them to be properly accounted for during the design stage of structure subjected to blast loads. 16

17 This research will: Develop a Single Degree of Freedom (SDOF) algorithm for axial and transverse loads on a reinforced concrete column and implement it within the Dynamic Structure Analysis Suite (DSAS) Version 2. (Center for Infrastructure and Physical Security, University of Florida (CIPPS, UF)). Model the columns in ABAQUS Version (Dassault Systèmes, 28) using the same physical and material properties. Validate the above software applications with available experimental data. Validate analytically between the two software applications. Conduct parametric study using DSAS Version 2. based on the above results 1.3 Research Significance The outcome of this research will verify the assumption stated in the problem statement. By knowing the significant effect of axial loads, building structures in high threat environment will have a better chance to resist and endure when subject to blast effects. On the other hand, for structure that may not be subject to these conditions, excluding the effect of variable axial loads will reduce the cost and the runtime required for the analysis and design. 17

18 CHAPTER 2 LITERATURE REVIEW 2.1 Introduction In the past few decades, there are numbers of books, publications, and technical papers explaining and discussing the effects of blast load on structures, particularly on the combined effects of flexure, axial, diagonal and direct shears (Biggs 1964, Murtha and Holland 1982, Baker et al. 1983, Krauthammer et al. 1988). Properties and behavior of blast when exerted on a structure were experimentally studied and recorded. Equations for calculations of all necessary parameters were either derived mathematically or empirically. The results were well explained and summarized in tables and charts allowing the users to expediently obtain pertinent information for using in the design of structures (Department of the Army, 199). Although fragmentation is normally associated with blast, it is not within the scope of this work. Therefore, the following sections in this chapter will provide a brief summary of these studies with respect to blast only as well as the required information to be used in this research. 2.2 Blast loading on Structures When explosive detonates, it generates a sudden, violent release of energy. After the arrival time, t A, following the detonation, the pressure peaks at its highest value, P so above the atmospheric pressure (1 ATM = 14.7 psi). There are two types of blast bursts with each type generates different effect. These are air-burst and ground-burst. If the charge is air-burst, depending of the angle of attack, the P so is further increased by the reflection off from the ground as shown in Fig If the charge is ground-burst, then the P so is at its maximum as the reflection of the blast wave occurs immediately as shown in Fig

19 Figure 2-1. Air-burst explosion (TM5-13, 199) Figure 2-2. Ground-burst explosion (TM5-13, 199) 19

20 As the shock wave moves through air, it is followed by an air pressure pocket which travels at a slower speed. This air pressure, a function of time, is known as the dynamic pressure q(t). As shown in Fig. 2-3, two stages are formed within a very short duration. An outward burst of the blast wave immediately raises the pressure over the ambient atmospheric pressure. This is also known as the positive phase. Shortly after that, this pressure drops below the ambient atmospheric pressure (negative phase) as the distance increases. P(t) Reflected Pressure Area i S P r Incident Pressure P s P 1 ATM P r - P s - t A t t - t T POS T NEG Figure 2-3. Typical blast wave pressure time-history graph Where: P so Incident pressure, Pr Reflected pressure, Is/W Scaled unit positive incident impulse, Ir/W Scaled unit positive normal reflected impulse, ta/w Scaled time of arrival of blast wave, tpos/w Scaled positive duration of positive phase, U Shock front velocity, L w /W 1/3 Scaled wavelength of positive phase. 2

21 A scaled-distance, Z, expressed in terms of stand-off distance, R G, and the Trinitrotoluene (TNT) equivalent charge weight, W, is used as a common factor to determine all the above parameters. The scaled-distance, Z, is calculated as follow: Z = R G 3 W (2-1) Depending on the category of explosion, air-burst (spherical) or ground-burst (hemispherical), data for the listed parameters can be obtained from the charts provided in the US Army TM 5-13 (199). Calculations for these parameters can be found in numerous references. Brode (1955) introduced the calculations for the over pressure in unit of bars: 6.7 P = 1 Z + so bar for P > 1 bars (2-2) 3 so P so = bars for.1 < P so < bars (2-3) Z Z Z This formula was later refined by Newmark and Hansen (1961) for ground-burst explosion: W W P 93 so = + 3 bars (2-4) 3 R R Other parameters such as shock front velocity, U, air density behind the shock front, ρ s, and dynamic pressure, q s, were introduced by Rankine and Hugoniot (187): U = 6 P so + 7 P 7 P a (2-5) ρ s = 6 P P so so + 7 P + 7 P ρ (2-6) q s = 2 ( P 5 P so 2 so + 7 P ) (2-7) 21

22 Where a is the speed of sound and ρ is the density of air beyond the blast wave at atmospheric pressure. In the case where reflection occurs, then: P r = 2 P so ( 7 P + 4 Pso ) ( 7 P + P ) so To compute the value of pressure, P, along the curve, one can use the Modified (2-8) Friedlander equation: t ( t α ) P( t) = P 1 e (2-9) max T pos The impulse, the area under the positive portion of the blast wave time history graph as shown in Fig. 2-3, can be calculated by integrating Equation 2-9 i s T = pos P( t) dt (2-1) However, approximation can be made for Fig. 2-3 by considering only the positive portion of the curve as a triangle. It can be simplified and re-drawn as shown in Fig. 2-4 P(t) P(t) P max i S P max i S t A T pos B T pos t Figure 2-4. Partial and simplified Time-History graph. A) partial blast wave, B) simplified blast wave 22

23 The relation of impulse with respect to a blast pressure and time can then be approximated based on Fig. 4B: i 1 2 s = Pmax T pos Since i s and P max can be determined from Equations 2-4 and 2-1, the T pos can be (2-11) computed by manipulating Equation Structure and Its Equivalent System Introduction For most structure, there will be an infinite number of degrees of freedom. It will be cumbersome and inefficient to analyze the structure in this form, not to mention the unfeasibility at times. Therefore, it is often possible to reduce the system to a single degree of freedom (SDOF) system to simplify the process. To achieve this, equivalent parameters for the SDOF system such as mass, M e, stiffness, K e, load, F e and load time function F(t) need to be setup. Although the response of the equivalent system, in term of forces and stress, are not the same as that of the actual system; the deformation and time, however, are the same in both systems. As such, selection of the equivalent system should be based on the criteria that the deflection of the concentrated mass is the same as that for a significant point on the actual structure as shown in Fig w(x) W M e u K e C L u A B Figure 2-5. Real system and typical equivalent SDOF system. A) Actual structural member. B) Equivalent SDOF system 23

24 2.3.2 Equivalent Mass As shown in Fig. 2-6, m(x) is the continuous mass of the element, u is the deflection at mid-span, u(x) represents the deflection along the element, and φ(x) is the assumed-shape function of the element. m(x) x u(x) L u ϕ (x) Figure 2-6. Mass diagram Assuming that the displacement, u, and the velocity, u, can be approximated as: u( x, t) = φ ( x) u( t) u ( x, t) = φ ( x) u ( t) (2-12) (2-13) Then the kinetic energy of the distributed mass system can be: 1 KE = 2 or 1 KE = 2 L L m( x) ( u ( x, t)) m( x) ( φ ( x)) 2 2 dx ( u ( t)) 2 dx (2-14) (2-15) And the kinetic energy of the equivalent system is KE e = 1 2 M e ( u ( t)) 2 (2-16) If M e is selected so that the kinetic energy of the real system is the same as that of the equivalent system, then by equating and simplifying the right-hand sides of Equations and 2-16, M e for the distributed mass system can be obtained and expressed as below: 24

25 M e = L 2 m( x) φ ( x) dx Similar process can be applied for the lumped mass system, which will yield: (2-17) M e n = [ m i= 1 i 2 φ ( x )] i (2-18) Thus the mass factor K M is the ratio of equivalent load, M e, to actual total load, M t. K = M M M e t (2-19) Equivalent Load Using the parameters shown in Fig. 2-7 as well as defining w(x) as the distributed load and P i as the concentrated load at location i, the equivalent load can also be determined following similar approach. P 1 P 2 P n w(x) x 1 x 2 u(x) u ϕ (x) x n L Figure 2-7. Load diagram The work done by the external load, WE, on the real system must be equal to the work done by the external load, WEe, on the equivalent system. Thus by defining, equating and simplifying the expressions for WE and WE e, the equivalent load, P e, for the distributed load system can be expressed as follow. 25

26 WE = L w( x, t) u( x, t) dx (2-2) Where u(x, t) is as previously defined, w(x, t) is expressed as below, and φ 2 (x) represents the assumed-shape function for the distributed load. Since this is a uniformly distributed-load case, φ 2 (x) = 1. The expression for w(x, t) becomes w ( x, t) = φ2 ( x) w( t) = w( t) (2-21) Therefore, WE = L w( t) φ( x) u( t) dx (2-22) Or WE = w( t) u( t) φ( x) dx L (2-23) WE e The work done by the equivalent external load on the equivalent system will be: = P u(t) e (2-24) Equating and modifying the right-hand sides of the expressions for the work done on the real system, WE, and the work done on the equivalent system, WE e, P e for the distributed load system can be obtained: P e = w( t) φ( x) dx L The same process can be repeated again for the concentrated load case; of which: (2-25) P e n = ( Pi φ( xi )) i= 1 Thus the load factor K L is the ratio of equivalent load, P e, to actual total load, P t. (2-26) K = L P e P t (2-27) 26

27 2.3.4 Shape Functions As shown in Section 2.3.4, to obtain the equivalent mass and equivalent load, it is required to use the assumed-shape functions φ and φ(x). Along with the derivation of the above factors, Biggs (1964) also provided some assumed-shape functions for simply supported beams and one-way slabs under various types of loading conditions. The data is, however, only applicable for either elastic or fully plastic and nothing in between. Employment of these assumed-shape functions will introduce some small errors and will be limited to the outlined load cases. Therefore, to correct these errors and eliminate these constraints, modifications to the employment of these shape functions will be required. Various approaches were done in the past with the attempt to overcome the above-mentioned limitations. Summary of some of these approaches will be provided in the later section Resistance Functions In general, when an external load is exerted on a column, the column tends to produce a resistance force trying to reinstate it to its original position. Biggs (1964) suggested, as shown in Fig. 2-8A, the three possible shapes corresponding to the three categories of materials: brittle, ductile, and plain concrete or instable structures. Simplification is made for most structures by using the bilinear functions in computing the resistant factor as shown in Fig. 2-8B; where R m is the maximum plastic-limit load that the beam could support statically. Thus in the linear elastic range, the resistance factor, K R, is the same as the load factor, K L, due the fact that the deflection is the same for both systems. Since this does not apply nonlinear situation, further development will be required and will be discussed in another section. 27

28 R Brittle Ductile R 1 R m Plain / Instable k A el B Figure 2-8. Resistance functions. A) Actual. B) Simplify (Biggs, 1964) The general equation of motion and its applicable form for a linear elastic system can be expressed as shown in Equations 2-28 and 2-29, repectively. m u + c u + k u = F(t) M e u + C u + K L R( t) = Pe ( t) (2-28) (2-29) Where R(t) is the resistance function that replaces the product of the spring stiffness and the displacement for an elastic beam and is defined as the restoring force in the spring, and the maximum resistance is the ultimate load the beam can carry under static conditions for an inelastic beam (Krauthammer et al., 1988). 2.4 Static Analysis Flexural Behavior Figure 2-9 shows a typical stress and strain diagram of a cross section in the plastic deformation state under axial load. The axial force, P, at any instance, t, can be computed from the blast pressure p(t) as outlined in Equation

29 ε cm σ c h d st d sb h/2 z ε z ε h/2 φ P b Strain, ε Stress, σ Figure 2-9. Stress and strain diagram of the cross section Moment-curvature relationship can be derived using the computed values of stresses and strains at any given point on the curve. Defining ε h/2 as the strain value at the mid-depth of the cross section and φ is the curvature at the corresponding point, the value of strain at any depth z, ε z can be calculated using similar triangle. Thus: ε z ε ( h 2) tan( φ) = ( h 2) z (2-3) Assuming small angle is used, the above equation can be re-written as: ε z h = ε ( h 2 ) + z φ (2-31) 2 From the constitutive relation of stress and strain, stress at any point on the cross section can be expressed as a function of strain at the corresponding point. σ z = f ( ε z ) (2-32) M The associated moment can then be defined as: h h = σ z z dz 2 (2-33) Equations 2-31 and 2-33 define the Moment-Curvature relationship for the cross-section. 29

30 2.4.2 Influence of Shear on Flexural Response Aside from flexure failure, shear failure is also a factor that needs to be considered. Two types of shear failures commonly known are diagonal shear and direct shear. Fig. 2-1 shows a typical shear failure of a column. Figure 2-1. Typical shear failure of a column (MacGregor and Wight, 29) To account for shear effect in the design for members that are subject to shear and flexure only, ACI uses Equation However, further knowledge is required to fully understand the actual behavior and the interaction between shear and flexure. V c ' = 2 f b d (2-34) c w Where: V c f' c the nominal shear strength. the compressive strength. Numerous studies were conducted to examine the influence of shear on flexural response (Kani, 1966, Placas and Regan, 1971, Haddadin et al., 1971, Bazant and Kim (1984), Ahamad 3

31 and Lue., 1987, Krauthammer et al., 1988). It was determined that the failure due to shear mainly depends on the reinforcement ratio, ρ, as well as the shear span-effective depth ratio, a /d; where a is the distance from the support to the load and d is depth taken from the top of concrete beam to the first layer of reinforcing bars. Figs and 2-12 show two models proposed by Kani (1966) and Ahamad et al. (1987) respectively; where M u /M fl is the ratio between the ultimate moment due to shear and flexure and the ultimate moment due to pure flexure. This ratio is known as the shear reduction factor (SRF). Figure Flexure-shear interaction model valley of diagonal failure (Kani, 1966) 31

32 Figure Flexure-shear interaction model (Ahmad et al., 1987) These two models were evaluated and further developed and modified by Krauthammer et al. (1988) and Russo et al., (1991 and 1997). Detailed discussion on the developments by these two authors is not part of the scope of this work and can be found in Krauthammer et al. (24). Based on the result found, approach by Krauthammer et al. (1988) concluded in the closer match with the experimental data presented by Kani (1966). Hence, this research paper will adopt this approach. This means that to take the shear effect into account more correctly, the moment will be multiplied by the SRF and the curvature will be divided by the SRF (Krauthammer et al., 24). This, in turn, produces a more accurate result for both moment and curvature Influence of Axial Force on Shear and Flexural Responses The presence of axial load enhances the moment capacity of the cross section as well as delays cracks from occurring (Krauthammer et al., 1988). This is because the compressive load increases the normal stress and reduces effect of the principal tensile stress. To maintain equilibrium, the sum of forces must still be zero. Thus: 32

33 Fx = h dz P = z σ (2-35) The current code, ACI 318-5, accounts for the axial load effect on flexure and shear by employing the equations for nominal shear strength, V c, and nominal flexural strength, M n. Equations 2-36 and 2-37 represent the lower and upper bound of the shear strength respectively: V c N u = A g f ' c b w d (2-36) V c 3.5 f N ' u c bw d 1+ (2-37) 5 Ag M n = M u N u (( 4 h) d ) 8 (2-38) Where: V c : nominal shear strength provided by concrete. N u : factored axial force normal to cross section. A g : section gross area. f c : specified concrete compressive strength. b w : width of the section. d: is the effective depth which is from the top compression fiber to the centroid of the longitudinal tension reinforcement. Mattock and Wang s studies (1984) conducted a study and found that Equations 2-36 and 2-37 were too conservative. The recommended replacements for Equations 2-36 and 2-37, respectively, are: ' 3 N u V = + c 2 f c 1 (2-39) ' Ag f c 33

34 V c 3.5 f ' c.3 N + A Direct Shear Mode of Response g u (2-4) There have not been many studies done on the direct shear effect. In addition to the above mentioned factors, Park and Paulay (1975) also suggested that the transferring of high shear stress across a weak section where cracks have formed was another factor that indicated the significant effects of direct shear on flexural members. The failure due to direct shear was further proven by the dynamic tests that were conducted by Kiger et al. (1984) and Slawson (1984). Hawkins (1974) proposed an empirical model on the behavior of shear stress versus slip; which did not include the effect of compression loads. This relationship was later modified by the Krauthammer et al. (1988) to include the effects of load reversals. Shear Stress, τ τ m B C K u τ L τ e A D E K e O δ 1 δ 2 δ 3 δ 4 δ max Slip, δ Figure Direct shear slip relationship (Krauthammer et al., 22) Summary of the slip range and the associated descriptions for each segment in Fig is shown in Table

35 Table 2-1. Definition of direct shear slip δ relationship (Hawkins, 1974) Slip Segment -3, δ x 1 Description in OA 4 Elastic response. Positive slope K = δ. e τ e ' τ m τ e = f c (2-41) 2 AB 4 12 Slope decreases but remains positive. Reaches maximum shear stress, τ m. ' ' τ m = 8 f c +.8 ρ vt f y.35 f c (2-42) Where ρ vt is the ratio between the area of reinforcement crossing the As shear plane and the gross area, ρ vt = A fy steel yield strength BC Maximum shear stress, τ m remains constant. CD 24 Constant decrease slope. Independent of reinforcement crossing the shear plane. ' K u = f c (2-43) DE > 24 Shear capacity remains constant. Deformation at E varies with the level of damage. Krauthammer et al. (22) went further to define the upper limits of segments CD and DE. Equations 2-42 and 2-43 show the upper limits of D and E respectively: g.85 As f τ L = A δ max g x e 1 = 6 ' s (2-44) (2-45) Where 9 x = 2.86 d f ' c b 35

36 Based on the proven fact that axial load increases the strength of the beam in a sense that it deters the cracks from extending into the compression block, it is safe to assume that the maximum stress will also increase. Hence, it is proposed that with the effect of axial load, the value of τ m (Equation 2-42) will increase its value by 1 + (Nu / (2*Ag)). Thus the equation of τ m can be re-written as below. This point still needs to be proven as part of this research. τ m = N ρ u ' ' 8 1+ f c +.8 vt f y.35 f c 2 A g 2.5 Dynamic Analysis (2-46) Newmark-Beta Method For a nonlinear system, a different approach will be required to obtain the solution for the equation of motion (Equation 2-28). There are numbers of methods that can be used to for computing the displacements, some of which are the acceleration method (implicit method) and the central difference method (explicit method). The main difference between these two is the time at which the equation of motion is satisfied. The former method satisfies the equation of motion at time t i+1 and the latter one is at time t i. For this research, the linear acceleration (implicit) method will be used. It is basically a special case of Newmark-Beta Method (Newmark and Rosenblueth, 1971), where γ = 1/2 and β = 1/6. A brief summary for this method is as follow: Knowing yi and y i at time t i and with Δt is the time step, from the equation of motion (Equation 2-12), compute for y i. Estimate y i+ 1. Compute y i+ 1, and y i+1 using the two equations below with the values of γ = 1/2, β = 1/6. y [ γ y + (1 y ] i+ 1 = yi + t i+ 1 γ ) i (2-47) 36

37 2 1 ( t) β y + β y yi+ = yi + t y 1 i + i+ 1 i (2-48) 2 Using the equation of motion, Equation 2-28, and the above computed values of y i+ 1 and y i+1, compute for y i+ 1. Check if convergence is satisfied. If so, move to the next time step; if not, use the newly computed value of y i+ 1 for the next iteration. To ensure stability and accuracy for the nonlinear system, Δt should be less than 3 T π, where T is the natural period of the system. A flowchart for this method will be presented in the next chapter Dynamic Resistance Functions The dynamic resistance responses of a structure depend on a number of factors. These include the stability nonlinearity, the geometric nonlinearity, and material nonlinearity of the structure. When an external load that acts on a structure is less than the yield load of the structure, the response of the structure is still in the elastic range. Thus, the resistance force or the internal force is the product of structure stiffness and the displacement (F int = R = k*u). However, when the external load is greater than the yield load, the resistance force or the internal force is no longer linear and becomes a function of displacement (F int = R = fn(u)). Figs 2-14 and 2-15 represent two models that were proposed by Clough et al. (1966) and Sozen (1974) that illustrated the nonlinear behaviors of a structure. Both of these models were portrayed by a number of bilinear resistance functions, of which each segment represented a loading stage. The order of loading sequence is as shown in alphabetical order. 37

38 M/M U 1 a b g h f o 1 c i / Y a, b, c,... is the loading sequence e d - 1 Figure Degrading stiffness model (Clough, 1966) Figure Idealized hysteresis loops for reinforced concrete (Sozen, 1974) Based on these two models, Krauthammer et al. (1988) proposed a piecewise multilinear-curve model (Fig. 2-16) that included all the above mentioned nonlinearities and portrayed both elastic and inelastic response of a beam with the following assumptions: 38

39 a. Beam is symmetrically reinforced. For unsymmetrical reinforced beam, two R-Δ curves will be required for both negative and negative loadings. b. Maximum displacement is reached during the first positive cycle, which is valid only for blast or impact loads. The elastic range exists when the maximum dynamic displacement (Δ max ) is less than the yield displacement (Δ y ). Within this range, the beam will behave elastically and oscillation will occur along line A-A. This oscillation will eventually come to rest once external damping dissipates all the energy. As Δ max becomes greater than Δ y, the beam will be in the inelastic range, at which point permanent deformation will take place. The order of points A, B, D, E, F, G, D, and E in the figure below (quadrants I and IV) describes the sequence of loading and unloading that eventually forms the plastic deformation (Δ plastic ) of the beam when all the energy is completely dissipated. Point C represents the flexure failure of the beam. Figure Typical response of a SDOF system (Krauthammer et al., 1988) 39

40 2.5.3 Modified Equivalent Parameters for SDOF System. As mentioned in Section 2.3, the mass factor and the load factor derived by Biggs (1964) were based on the assumed-shape functions that were either for elastic or fully plastic and nothing in between. Moreover, these assumed-shape functions were also derived for specific cases of end supports and loading conditions. With a better insight on the behaviors of a structure member in both elastic and inelastic range based on the above model proposed by Krauthammer et al. (1988), these factors can be re-evaluated to account for the behaviors of the member in the transition stage between elastic and inelastic. The procedures to compute mass factor and load factor were extracted from the above mentioned reference Mass factor The mass factor can now be computed along the load-deflection curve by taking the integration over the length of the beam for each load step j. Using Fig. 2-6 and all previously defined variables, the equivalent mass and the mass factor at load step j can be expressed as below. M L 2 e = m x) φ ( x j ) dx + j i= 1 n 2 [ ( mi ) ( xi ) j ] j ( φ (2-49) K M = j M M e j t (2-5) Thus the mass factor, K M, for each time step can be computed: K M = K M j KM + ( J + 1) ( j+ 1) K M j j ( ) j (2-51) Where Δ j < Δ Δ (j+1). Reference to Fig. 2-16, KM is computed for every time step until it reaches point B, which is the maximum inelastic displacement. Beyond that, it will remain constant based on the 4

41 assumption that the shape functions do not significantly change after the formation of the plastic hinges Load factor The procedure to re-evaluate the load factor for the transition stage between elastic and plastic is similar to that of the mass factor. Hence, using Fig. 2-7, for each load step j with shape function φ(x) (j,i) where i is the location to be evaluated, the load factor for each load step can be expressed as: K L j = φ x ( ) ( j, i) (2-52) Thus the load factor for each time step will be K L = K L j K + L ( J + 1) ( j+ 1) K L j j ( j ) (2-53) Where Δ j < Δ Δ (j+1) Dynamic Reactions Biggs (1964) provided a series of tables of equations for the dynamic reactions of various loading scenarios with different boundary conditions. As mentioned earlier, the limitations of these equations are that they are only applicable to either perfectly elastic or plastic structures and are bounded by specific load cases. Computations of the dynamic reactions based on these equations will introduce inaccurate results for the purpose of this paper. Therefore, another approach is required. Krauthammer et al. (1988) introduced a procedure that could be used to neutralize the above limitations. This procedure was based on the assumption that the distribution of the inertia forces is identical to the deformed shape function of the beam as shown in Fig

42 γ ' 1i Q(t) γ ' 2i M 1 M 2 γ 1 Q(t) Ψ (x) γ 2 Q(t) Figure Dynamic reactions for beam with arbitrary boundary conditions (Krauthammer et al., 1988) Summary of the procedure is as follow (Krauthammer et al., 1988): a. For each load step i, obtaining the reactions at each end of the of the element and compute the corresponding load proportionality factor γ: γ Q / Q (2-54) 1 i = 1i i γ Q / Q (2-55) 2 i = 2i i Where: Q 1i and Q 2i are the static reactions at load step i of end 1 and 2. Qi is the load at step i. γ1i and γ 2i are the load proportionality factors at end 1 and 2. b. For every load step, compute the Inertial Load Factor, IFL: 1 L ILFi = ( ( xi )) dx L ψ (2-56) Where: IFL is the load factor associated with the distribution of the inertial forces. ψ(x)i is the deflected shape function at load step i. c. Compute the inertia proportionality factors γ 1i and γ 2i at every load step i. These factors are approximated using the principles of the linear beam theory. d. Compute the dynamic reaction at each end of the element for each time step: 42

43 V V ( γ Q( t ) + ( ILF γ M X ) 1 = 1i ) 1i ( Q( t ) + ( ILF γ M X ) 2 = 2i ) 2i t t (2-57) γ (2-58) Where: Q(t) is the forcing function. M t is the mass of the beam. X is the acceleration. γ 1i, γ 2i, γ 1 i and γ 2i specific displacement at every time step by using the following linear interpolation equation: γ i+ 1 γ i γ = γ i + ( i ) for < < i 1 i+ 1 + i i (2-59) Δ is the dynamic displacement at the specific time step. γ is the generic name for inertia and load proportionality factors. 2.6 Pressure-Impulse (P-I) Diagrams Characteristics of P-I Diagrams P-I diagrams are graphical tools used to determine the potential damage of a structure caused by dynamic loads. Detailed descriptions on P-I diagrams can be found in numerous references (Krauthammer, 28). There are three distinguished regions on the P-I curve, as shown in Fig These are the Impulsive Loading Region, the Quasi-Static Loading Region and the Dynamic Loading Region. In addition, there are two asymptotes. The Impulse Asymptote is tangent to the Impulsive Loading Region and the Pressure Asymptote is tangent to the Quasi-Static Loading Region. In the Impulsive Loading Region, the response time of the structure is much longer than the duration of the loading. Hence, before the structure can experience any permanent deformation, the load is already dissipated. In the Dynamic Loading Region, the duration for both loading and natural period is approximately the same. The 43

44 response of the structure in this region depends on the loading history. In the Quasi-Static Loading Region, the loading duration is much longer than the natural period. Therefore, the structure experiences maximum deformation before the load completely dissolves. (Smith and Hetherington, 1994) 5 Impulsive Loading Region 4 3 Pressure, psi 2 1 Dynamic Loading Region Quasi-Static Loading Region Pressure Asymptote Impulse Asymptote Impulse, psi-msec Figure Typical pressure-impulse diagram Derivation of P-I Diagrams Although approach to develop a P-I diagram for a complex non-linear structure could be possible, it would be, however, extremely cumbersome. Hence, numerical approach should be used. The P-I diagram is derived from the results of numerous single dynamic analyses, where the computed threshold points are used to plot the P-I curve. Since the process to obtain these threshold points is intensive in term of computational time, an effective and efficient search algorithm is required. 44

45 Blasko et al. (27) developed a good search engine where a single radial search direction was originated from an arbitrary pivot point that was located in the failure zone of the P-I diagram as shown in Fig The iteration process continued where another arbitrary point between the point in the safe zone and the first assumed point was evaluated. The same procedure was repeated until all the threshold points were successfully acquired. This process can be completed for structure that may also experience more than one failure modes such as shear and flexure. The results from both of these failure modes can then be plotted together for use in evaluating the structure to determine the likelihood of the mode of failure as shown in Fig Figure Search algorithm in developing P-I diagram (Blasko et al., 27) 45

46 Pressure Failure in Mode 2 Failure in Mode 1 & 2 Mode 2 (Direct Shear) Safe Failure in Mode 1 Mode 1 (Flexure) Impulse Figure 2-2. Typical P-I diagram for multi failure modes (Chee, 28) 2.7 Summary In this chapter, the property of explosive blast and the behavior of a beam under the influence of transverse and axial loads due to blast pressure were briefly reviewed along with the possible failure modes due to shear and flexure. The effect of axial load on these modes of failures was also considered. Moreover, the transformation from an actual structure to an equivalent SDOF system through the use of equivalent parameters such as mass factor, load factor and the dynamic reactions that were based on the employment of the assumed-shaped functions was discussed. Since this problem includes nonlinearity, closed-form approach would be very cumbersome and inefficient. Hence, direct integration techniques both implicit and explicit were considered. A short summary on the Pressure-Impulse diagram and how it can be used to quickly determine the failure of a structure was provided. The brief discussions on each topic in this chapter provide adequate source of information to form a basis for the analysis that will be discussed in the next chapter. 46

47 CHAPTER 3 METHODOLOGY 3.1 Introduction This chapter provides an overview of the structure being considered as well as outlines the approaches used in determining the effects of variable and constant axial loads on a reinforced concrete column that is also subjected to transverse loads. Assumptions and simplifications used in the implementation steps will also be included in the appropriate sections of this chapter Overview of Structure 3.2 Load Determination In either case of air or ground-blast as shown in Figs. 2-1 or 2-2, once the first wave of blast strikes the building, it will destroy architectural items such as windows and doors. This creates openings in the structure allowing subsequent blast waves to act as internal pressures in the outwards and upwards directions against walls, floor and roof of the building. The structure considered in this case is as shown in Fig. 3-1, of which only the column is being investigated. It should be noted that the loading diagram caused by both exterior and interior pressure is much more complicated than shown. For simplicity purpose, only loads that have significant effects on the column are being considered. In addition, although the arrival time, t A, of the blast load will be different for the column and the supported mass; for ease of computations, it is assumed that t A is the same for the above mentioned structures. Moreover, to reduce the computer runtime, the end boundary conditions for the column will be reduced to simply support condition rather than fixed-fixed condition. 47

48 P d (t) P u (t) F(t) H L Figure 3-1. Blast loads on structure The column of the above structure will be subject to flexure, diagonal shear and direct shear induced by: a. Lateral dynamic loads F(t). b. Downward loads, P d (t). c. Upward loads, P (t). u The equivalent system for the above structure will be a multi-degree of freedom (MDOF) system under the influence of transverse loads and axial loads. Solving for the above structure using multi-degree of freedom approach may be inept. Thus the system will be de-coupled into two independent members that will eventually be reduced to two equivalent single degree of freedom systems. The first member to be considered is the supported mass. As shown in Fig. 3-2, P(t) is the net load resulting from the differential pressure between upward and downward pressure. R1(t) and R 2 (t) are the dynamic reactions, M is the internal moment, and Δ is the deflection due to load. 48

49 P(t) P(t) L A Β M L? M Figure 3-2. Load diagram for supported mass. A) Load diagram, B) Free-body diagram The second member to be considered is the column; which is the main focus of this research. Loads that act on this member include the transverse load F(t), the internal moment M, the weight of the structure above, the variable axial loads which are the dynamic reactions resulting from the first member, R(t), and the self-weight of the column. It should be noted that in both cases, members are subject to transverse loads that could induce failure by either shear or flexure. Thus, consideration of the effects of transverse load is required. R(t) R(t) M Plastic Hinge 3 Locations (Typ) δ F(t) H F(t) H A B M Figure 3-3. Load diagram for column. A) Initial stage, B) Deformed stage 49

50 3.2.2 Effect of Transverse Loads. Previous studies on box-type buried reinforced concrete structure subject to blast load (Kiger et al., 1984, Slawson et al., 1984, and Ross, T.J and 1985) indicated that when element of the studied structure failed due to shear, the flexural response was negligible. On the other hand, when element of the studied structure failed due to flexure, the element was able to withstand the shear forces at the early stage. Based on the result, Krauthammer et al. (1988) suggested the employment of two separate SDOF systems for evaluating flexural and direct shear responses of a beam. Verification against the failure criterion of the computed results from these two SDOFs was conducted at the end of each time step to determine the mode of failure. As mentioned above, the failure of the two members could also be induced by either flexure or direct shear. Therefore, this research paper will employ the approach suggested by Krauthammer et al. (1988). This means that each of the two members described above will have two SDOF systems. The failure mode of the first member will be used as a governing factor in the computation for the second member. In other word, the dynamic reactions resulting from the failure mode in the first member, by direct shear or flexure, will be used as the variable axial loads acting on the second member Axial Loads As indicated above, two types of axial loads will be used in the modeling and the required computations. These are constant and variable axial loads. The constant axial loads are assumed to be induced by the weight of the supported mass over the period of time. Application of various magnitudes of constant axial loads to the column will be considered for comparison purpose. Variable axial loads are derived from the computations of the dynamic reactions caused by the effect of the blast load on the supported mass. 5

51 3.3 Load Deformation Analysis Within the elastic range, the stress distribution of a cross section at any point along the beam will remain linear. As the load increases, the resulting moment also increases until it passes the yield point; when the stress distribution of the cross section becomes nonlinear and a plastic hinge is formed. Since the load-deformation relationship is no longer linear, equations derived for the linear elastic are no longer valid. As such, a different method is required to trace the nonlinear path of the beam behavior. One of the most commonly used methods was the Newton-Rhapson method (Fig. 3-4); where for any load-displacement function, the displacement value at point B could be determined based on the known value at point A. The aim of each iteration, i, was to reduce the out-ofbalance load ΔR i or the displacement Δu i to a satisfactorily small value. Thus: = K B( i u 1) i R ( i 1) (3-1) Where: i the iteration number. K tangent stiffness matrix. Δu i incremental displacement at i th iteration. ΔR(i-1) incremental load at (i-1) th iteration. Load R B R B - F B A K B 1 R B - F B A 1 K B 1 1 R A u 1 u 2 u A u 1 u 2 u B Displacement, u Figure 3-4. Newton-Rhapson method (Bathe, 1996) 51

52 It was, however, found that this method did not converge for zero slopes. Thus for typical load-deformation diagram as shown in Fig. 3-5, this method would not yield the best result as it might not be able to pass the post-buckling response point. Load Postbuckling response Load decreases Small load increments Large load increments Displacement Figure 3-5. Typical load-deformation diagram of a structure (Bathe, 1996) To account for this issue and to allow for the post-buckling response of a structure, the Cylindrical Arc-Length method developed by Crisfield M.A. (1981) is used. This procedure allows the tracing to be possible even when the slope of load-deflection curve is negative. The main difference between this method and the Newton-Rhapson method is the assumption that the load vector, R, varies proportionally during the response calculation and the use of the load multiplier; which will need to be determined. In short, Newton-Rhapson is a forced-controlled method; whereas, Cylindrical Arc-Length is a displacement-controlled method. Using Fig. 3-6, the algorithm of this procedure can be described as follow with detailed discussion can be found in the reference (Cook et al., 1974): 52

53 Load λ R B λ R A l Displacement, u Figure 3-6. Spherical constant arc length criterion for SDOF system (Cook et al., 1974) a. The governing finite element equations for n equations in (n+1) unknowns λ K ( t+ t ) R F( t+ t) = τ Or u i = {[ λ ] } [ ] ( t+ t) + λ ( 1) i R F( t+ t) ( 1) i i (3-2) (3-3) Where: λ (t+δt) is the load multiplier at time (t+δt) that needs to be determine and can be increased or decreased. R is the reference load vector for n DOFs of the FEA model. It can contain any loading on the structure but is constant throughout the response calculation. F (t+δt) is the vector of n nodal point forces corresponding to the element stresses at time (t+δt). i is the iteration order. K τ is the coefficient matrix. b. Additional equation requires for determining the unknowns vector of displacement increments, Δu, and load multiplier increment, Δλ is a constraint equation between Δu i and Δλ i. 53

54 f ( λ i, u ) = i (3-4) u i Where: [ u( t+ t) ] ut = i (3-5) λ i [ λ t+ t) ] t = ( λ (3-6) i c. Effective constraint equation is given by the spherical constant arc length criterion T ( u ) u 2 2 i i ( λi ) + = l β (3-7) Where u i is the total increment of displacement within each load step for the i th iteration. λi is the total increment in the load multiplier for the i th iteration. l is the arc length for the time step. β is the normalizing factor. 3.4 Computations of Dynamic Reactions, Shear and Flexural Responses As mentioned in Sect , two separate SDOF systems will be used for the supported mass and the column respectively. Within the SDOF system for the supported mass, a sub-set of two SDOF systems will be used to determine the mode of failure. The governing dynamic reactions of the supported mass produced either by shear or flexure will be used as the timedependant axial loads applied to the column Computation of Dynamic Reactions for the Supported Mass The equivalent SDOF system and the associated free-body diagram, as shown in Fig. 3-7, can be used to illustrate all the pertinent loads used in the computations. The complete process in determining the mode of failure of a structure as well as obtaining the required parameters such as acceleration, velocity, displacement and resistance associated with the failure mode is shown in Fig 3-9. Figure 3-1 presents the Newmark-Beta method, which is a part of the 54

55 complete process. Both of Figs 3-9 and 3-1 are applicable to Section and Section The results obtained from the complete process, particularly the dynamic reactions, will be used for the necessary computations in the next section P(t) P(t) K A M Ae C y M Ae *y" A B R A (t) C*y' Figure 3-7. Supported mass diagram. A) SDOF for supported mass. B) Free-body diagram Computation of Shear and Flexural Responses on Column. The column is subjected to three different types of loads. Firstly, it is experienced by a transverse load that could cause failure in either flexure or shear. Secondly, it is exerted by the dead load of the structure directly above it as well as its own weight. These are considered to be the static loads. Finally, it is also experienced by the variable axial loads which are the dynamic reactions resulted from the supported mass being subjected to the blast load. x R A (t) K F R F (t) C M Fe F(t) C*x' M Fe *x" F(t) A B Figure 3-8. Column with axial load. A) Equivalent SDOF system. B) Free-body diagram 55

56 START Input Parameters, Pertinent Data Initialize All Variables Compute Velocity, Displacement, Acceleration, Mass and Load Factors, Resistance No Initial Time Increment by Δt Last Time Step? Compute Applied Force (for beam and column) Convergence? yes Process to compute listed variables using Newmark- Beta Method Compute Plastic Displacement Flexural Failure? yes No Direct Shear Analysis? Compute Shear Force Compute Shear Velocity, Displacement, Shear Resistance and Acceleration No Convergence? No Process to compute listed variables using Newmark- Beta Method Shear Failure? yes No Outputs STOP Figure 3-9. Flow chart for determining dynamic reactions based on failure mode 56

57 START Known Initial Conditions: u,u Select Time Step Δh Compute u 2 u + ξωu + ω u = F( t, u, u) / m 2 Start Counter Increment Counter Counter - 1 Yes Estimate u i ( +1) u trial u ( i +1) u ( 1 2 β ) u u( 1) ) h i + β i ( i 1) = ui + u i h u ( i 1) = u i + ( u i + u + ( i+ 1) ) ( h 2) F e, M e, R e u 2 { F[ t( i+ 1), u( i+ 1), u ( i+ 1) ] M e} ( Re M e ) ξ ω u ( i+ 1) ω u( + 1) ( i+ 1) = 2 i No u u ( i+ 1) trial Yes Output u, u, u STOP Figure 3-1. Newmark-Beta method for computing displacement 57

58 3.5 Summary This chapter provided the layout of the structure as well as outlined the assumptions and limitations applied to the column being considered. It described the loads exerted on each member of the structure and the computation process to transfer the applicable loads from one member to another. The effect of transverse load on the column was also discussed. Approach on load-deformation analysis was outlined along with the approach on the computations of the dynamic reactions, shear and flexural response of the equivalent SDOF system. 58

59 CHAPTER 4 ANALYSIS 4.1 Introduction Numerous steps were taken in order to confirm the objective. Firstly, validations for the ABAQUS Version (Dassault Systèmes, 28) and the Dynamic Structure Analysis Suite (DSAS) Version 2. (CIPPS, 28)) were required to ensure that they could produce reliable results. This was completed by comparing the results obtaining from these two applications against known data of a series of experiment on reinforced concrete beams. Secondly, the validation was completed on a reinforced concrete column. However, since there was no experimental data available for the column, these two software applications were validated analytically using a standard size column with the same material properties of the experimental beam. Lastly, a series of columns of the same physical and material properties but different reinforcements configurations was arbitrarily picked from the Concrete Reinforcing Steel Institute Design Handbook (CRSI, 22) for further analysis in the parametric study to determine the effect of axial loads on the P-I relationships. This was followed by the analysis on the effects of variable axial loads on the columns. The computer code DSAS was modified to address the effects of axial force on RC columns, and it was used to derive the corresponding P-I curves. 4.2 Description of DSAS DSAS (CIPPS 28) is a comprehensive software suite developed for the analysis and assessment of structural members subject to severe dynamic loads such as blast and impact. The primary analysis engine in DSAS is based on an advanced single degree of freedom (SDOF) formulation and is capable of developing fully non-linear resistance functions for reinforced concrete, steel, masonry and other members with diverse end conditions using force or displacement-controlled solution procedures (Krauthammer et al., 199 and 23, DSAS User 59

60 Manual, 28). The moment curvature relationships for RC that are derived by DSAS are based on layered section analysis with fully nonlinear material models for steel and confined and/or unconfined concrete. The resistance function is based on a displacement controlled solution approach, and the Direct Shear function uses the Hawkins model. The present study enabled the development of an enhanced version of DSAS that allows for constant gravity loads to be specified and modifications can be made to account for dynamic variations in axial force. Moreover, DSAS is also capable to conduct Physics-based P-I analysis and produce the P-I diagrams. 4.3 Validations with Experimental Data Experimental Data Material and physical properties Data of five beams from the experiment conducted by Feldman and Siess (1958) were used as part of the validation. Detailed layouts of steel reinforcements for all beams are shown in Fig The main difference in the physical configuration between beam 1-C and the other four beams was the transverse reinforcements. Beam 1-C used opened stirrups and the other two beams used closed stirrups. Other properties of the beams are described in Tables 4-1 and 4-2. Figure 4-1. Beam 1-C layout (Feldman and Siess, 1958) 6

61 Modulus of Elasticity, Table 4-1. Concrete material properties b = 6 in, h = 12 in, d = 1 in, d = 2. in, span =16 in Beam Compressive Strength, Rupture Strength, f r, f c, ksi Ec, ksi ksi 1-C G H I J Table 4-2. Steel reinforcements material properties Tension reinforcement = 2 #7 bars, compression reinforcement = 2 #6 bars, stirrups = 16 #3 bars at 7 in on center Beam Compression reinforcements Tension reinforcements f y, Ksi E s, Ksi ε' y, in/in ε' sh, in/in fy, Ksi Es, Ksi εy, in/in εsh, in/in 1-C G H I J Loading functions Loading functions for all beams were reproduced by extracting the points from the loading graphs provided in the experimental report. For beam 1-G, due to the failure of test recording equipment, the actual loads were not properly recorded. Hence, it had to be estimated. Two main criteria were based on when conducting this process. Firstly, the load profile was assumed to follow the shape of the sum of the reactions curve. Secondly, it must take into account the inertia effect of the beam under loading. Loading for beam 1-G was stopped at.72 sec when the wooden stop was hit. For beam 1-J, it should be noted that it was subjected to two separate sets of impact loads. Similar to the loading situation for beam 1-G, the first set of loads stopped at.72 sec and resulted in a displacement of.5 in. It was then subjected to the second set of loads which was stopped at approximately.68 sec. For simplicity, these two sets of loads were combined in one simulation. Figs 4-2 to 4-6 show the loading functions for the beams. 61

62 Load (kips) Time (msec) Figure 4-2. Load function for beam 1-C (Feldman et al., 1958) Load (kips) Loads, Kips (Actual) Loads, Kips (Estimated) Time (msec) Figure 4-3. Load function for beam 1-G (Feldman et al., 1958) 62

63 Load (kips) Time (msec) Figure 4-4. Load function for beam 1-H (Feldman et al., 1958) 4 3 Load (kips) 2 Stop at 72 msec Time (msec) Figure 4-5. Load function for beam 1-I (Feldman et al., 1958) 63

64 First 72 msec Load (kips) 25 2 First 68 msec Time (msec) Figure 4-6. Load function for beam 1-J (Feldman et al., 1958) ABAQUS Validations All five beams were modeled with ABAQUS Version (Dassault Systèmes, 28). Although various mesh sizes were evaluated, it was found that a cubical mesh size of 1 inch yielded the most effective and economical results in terms of computer runtime and the accuracy of the outcomes. Hence, this mesh size and the solid element type C3D8I were used in the modeling of all the beams. Various types of material models in ABAQUS were also explored for the modeling of the concrete beam. It was determined that the Modified Drucker-Prager/Cap Model was the most suitable one to be used in this case for numerous reasons. Firstly, it could respond to large stress reversals in the cap region. Secondly, it provided the required inelastic hardening mechanism to account for the plastic compactions as well as the necessary controlling of the material expansion when yielding in shear (ABAQUS Analysis User s Manual). With 64

65 regards to the longitudinal and transverse reinforcements, two different element types were used. Beam element, type B31, was used to model the compression reinforcements. Surface element, type SFM3D4R, was used for the modeling of tension and transverse reinforcements. This was done because the surface-element type required less work to model compared to the beamelement type while still yielding the same result. Bond slip between concrete and steel reinforcements was taken into account using the built-in capability of ABAQUS known as the embedded region constraint. Fig. 4-7 shows a typical ABAQUS (Dassault Systèmes, 28) model for all the above-mentioned beams. Figure 4-7. Typical modeling of experimental beams using ABAQUS Six parameters were required for the Modified Drucker-Prager/Cap Model. These were the material cohesion (d), the angle of friction (β), the cap eccentricity (R), the initial yield pppp surface position (α), the transition surface radius (εε vvvvvv ), and the flow stress ratio (K). Fig

66 provides the graphical illustration of these parameters. Detailed explanations and derivations of these parameters can be found in the Section of the ABAQUS Analysis User s Manual, ABAQUS Version Figure 4-8. Graphical presentation of parameters in the Modified Drucker-Prager/Cap Model (ABAQUS analysis user s manual) Default values for most of the parameters were used with the exception of two parameters that had significant effects on the results. These were the material cohesion and the angle of friction. Computation of the material cohesion was based on the following procedure: ff cc = νν ff cc (4-1) νν =.7 ff cc 2 (4-2) dd = ff cc 4 (4-3) With regards to the angle of friction, for normal concrete, it should be taken about 37 degree. However, it was found that this value of the angle of friction and those values produced from Equations 4-1 to 4-3 for the material cohesion only provided a good starting point. The calculated values of these two parameters for each beam still required adjustments in order to 66

67 obtain the desired results. Numerous trials were made with different values of material cohesion and angle of friction in attempt to produce the outcomes that were comparable to the experimental results. Table 4-3 provides the summary of the best values of the parameters used for all five beams. Table 4-3. Material model properties for beams in ABAQUS Beam Material Cohesion, d Angle of Friction, β Cap Eccentricity, R Initial Yd Surf Pos, pppp εε vvvvvv Trans Surf Rad, pppp εε vvvvvv Flow Stress Ratio, K 1-C G H I J With regards to the material property of steel, Hsu theory (Hsu, 1993) on stress-strain relationship of mild steel was applied taking into account the strain rate hardening for both concrete and steel reinforcements. Detailed explanations on the computational procedure can be found in Chapter 7 of the above mentioned reference. The strain rate hardening and the enhancement factors for all five beams, which were required for the computations, were obtained from the previous research (Shanaa, 1991), and were implemented in the calculations of the stress-strain relationships of concrete and steel reinforcements. Using the data provided in Table 4-4, the stress and strain relationships for all five beams were calculated. The results are illustrated in Figs. 4-9 to 4-18 respectively. Table 4-4. Strain rate hardening and material enhancement factors (Shanaa et al., 1991) Beam Strain rate, Strain rate enhancement factors in/in/sec Compression - Reinforcements Tension - Reinforcements Compression - Concrete Tension - Concrete 1-C G H I J

68 75 6 Stress (ksi) Strain (in/in) Figure 4-9. Stress-strain relationship tension reinforcements beam 1-C 75 6 Stress, (ksi) Strain, (in/in) Figure 4-1. Stress-strain relationship compression reinforcements beam 1-C 68

69 75 6 Stress (ksi) Strain (in/in) Figure Stress-strain relationship tension reinforcements beam 1-G 75 6 Stress (ksi) Strain (in/in) Figure Stress-strain relationship compression reinforcements beam 1-G 69

70 75 6 Stress (ksi) Strain (in/in) Figure Stress-strain relationship tension reinforcements beam 1-H 75 6 Stress (ksi) Strain (in/in) Figure Stress-strain relationship compression reinforcements beam 1-H 7

71 75 6 Stress (ksi) Strain (in/in) Figure Stress-strain relationship tension reinforcements beam 1-I 75 6 Stress (ksi) Strain (in/in) Figure Stress-strain relationship compression reinforcements beam 1-I 71

72 75 6 Stress (ksi) Strain (in/in) Figure Stress-strain relationship tension reinforcements beam 1-J 75 6 Stress (ksi) Strain (in/in) Figure Stress-strain relationship compression reinforcements beam 1-J 72

73 Since ABAQUS Explicit required the use of true stress and logarithmic strain, the above calculated stress-strain values were then converted using the two equations provided in the ABAQUS manual (28) prior to input the data into the ABAQUS program: σσ tttttttt = σσ nnnnnn (1 + εε nnnnnn ) (4-4) εε llll pppp = ln(1 + εε nnnnnn ) σσ tttttttt EE ss (4-5) DSAS Validations Validation using DSAS was a much simpler task. This software application required the inputs of physical and material properties of the beam as well as load time history. Strain hardening could either be applied manually to the material property prior to input or through the built-in switch. For this validation, an average strain rate of.3 was used for all beams. Figure Typical DSAS data entry screen 73